Transportation problems

The table shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to each of three sales points, P, Q and R. It also shows the stock held at each supply point and the amount required at each sales point. A minimum cost solution is required.

The north-west corner method gives the following initial solution.

Taking AQ as the entering cell, use the stepping stone method to find an improved solution. Make your route clear.

Perform one further iteration of the stepping stone method to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, route, entering cell and exiting cell.

Determine whether your current solution is optimal. Justify your answer.

State the cost of the solution you found in (b).

Formulate this problem as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.

Three warehouses, P, Q and R, supply washing machines to four retailers, A, B, C and D. The table gives the cost, in pounds, of transporting a washing machine from each warehouse to each retailer. It also shows the number of washing machines held at each warehouse and the number of washing machines required by each retailer. The total cost of transportation is to be minimised.

Formulate this transportation problem as a linear programming problem. You must define your decision variables and make the objective function and constraints clear. You do not need to solve this problem.

The table shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to each of three demand points, 1, 2 and 3. It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required.

Use the north-west corner method to obtain an initial solution.

Taking D1 as the entering cell, use the stepping stone method to find an improved solution. Make your route clear.

Perform one further iteration of the stepping stone method to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, route, entering cell and exiting cell.

Determine whether your current solution is optimal, giving a reason for your answer.

The table below shows the cost of transporting one unit of stock from each of four supply points, 1, 2, 3 and 4, to each of three demand points, A, B and C. It also shows the stock held at each supply point and the stock required at each demand point. A minimal cost solution is required.

Add a dummy demand point and appropriate values to Table 1 in the answer book.

Table 2 shows an initial solution given by the north-west corner method. Table 3 shows some of the improvement indices for this solution.

Calculate the shadow costs and the missing improvement indices and enter them into Table 3 in the answer book.

Taking the most negative improvement index to indicate the entering square, use the stepping-stone method once to obtain an improved solution. You must make your route clear and state your entering cell and exiting cell.

Jameson cars are made in two factories A and B. Sales have been made at the two main showrooms in London and Edinburgh. Cars are to be transported from the factories to the showrooms. The table below shows the cost, in pounds, of transporting one car from each factory to each showroom. It also shows the number of cars available at each factory and the number required at each showroom.

It is decided to use the transportation algorithm to obtain a minimal cost solution.

Explain why it is necessary to add a dummy demand point.

Complete table 1 in the answer booklet.

Use the north-west corner rule to obtain a possible pattern of distribution.

Taking the most negative improvement index to indicate the entering square, use the stepping-stone method to obtain an optimal solution. You must make your shadow costs and improvement indices clear and demonstrate that your solution is optimal.

State the cost of your optimal solution.